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CSE 573 : Artificial Intelligence

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CSE 573: Artificial Intelligence

Reinforcement Learning II

Dan Weld

Many slides adapted from either Alan Fern, Dan Klein, Stuart Russell, Luke Zettlemoyer or Andrew Moore

1

Today’s Outline

- Review Reinforcement Learning
- Review MDPs
- New MDP Algorithm: Q-value iteration

- Review Q-learning
- Large MDPs
- Linear function approximation
- Policy gradient

Applications

- Robotic control
- helicopter maneuvering, autonomous vehicles
- Mars rover - path planning, oversubscription planning
- elevator planning

- Game playing - backgammon, tetris, checkers
- Neuroscience
- Computational Finance, Sequential Auctions
- Assisting elderly in simple tasks
- Spoken dialog management
- Communication Networks – switching, routing, flow control
- War planning, evacuation planning

Demos

- http://inst.eecs.berkeley.edu/~ee128/fa11/videos.html

Agent Assets

Transition & Reward Model

Simulator (w/ teleport)

World

Value Iteration

Policy Iteration

Monte Carlo

Planning

Reinforcement

Learning

Small vs. Huge MDPs

- First cover RL methods for small MDPs
- Number of states and actions is reasonably small
- Eg can represent policy as explicit table

- These algorithms will inspire more advanced methods

- Number of states and actions is reasonably small
- Later we will cover algorithms for huge MDPs
- Function Approximation Methods
- Policy Gradient Methods
- Least-Squares Policy Iteration

Passive vs. Active learning

- Passive learning
- The agent has a fixed policy and tries to learn the utilities of states by observing the world go by
- Analogous to policy evaluation
- Often serves as a component of active learning algorithms
- Often inspires active learning algorithms

- Active learning
- The agent attempts to find an optimal (or at least good) policy by acting in the world
- Analogous to solving the underlying MDP, but without first being given the MDP model

Model-Based vs. Model-Free RL

- Model-based approach to RL:
- learn the MDP model, or an approximation of it
- use it for policy evaluation or to find the optimal policy

- Model-free approach to RL:
- derive optimal policy w/o explicitly learning the model
- useful when model is difficult to represent and/or learn

- We will consider both types of approaches

Comparison

- Model-based approaches:
Learn T + R

|S|2|A| + |S||A| parameters (40,400)

- Model-free approach:
Learn Q

|S||A| parameters (400)

Supposing 100 states, 4 actions…

RL Dimensions

TD Learning

Q Learning

Optimistic

Explore / RMax

ADP e-greedy

Active

ADP

TD Learning

Direct Estimation

Passive

Uses

Model

Model

Free

Recap: MDPs

a

s, a

s,a,s’

s’

- Markov decision processes:
- States S
- Actions A
- TransitionsT(s,a,s’) aka P(s’|s,a)
- Rewards R(s,a,s’) (and discount )
- Start state s0 (or distribution P0)

- Algorithms
- Value Iteration
- Q-value iteration

- Quantities:
- Policy = map from states to actions
- Utility = sum of discounted future rewards
- Q-Value = expected utility from a q-state
- Ie. from a state/action pair

Andrey Markov

(1856-1922)

Bellman Equations

Q*(a, s) =

Value Iteration

- Regular Value iteration: find successive approx optimal values
- Start with V0*(s) = 0
- Given Vi*, calculate the values for all states for depth i+1:

Qi+1(s,a)

- Storing Q-values is more useful!
- Start with Q0*(s,a) = 0
- Given Qi*, calculate the q-values for all q-states for depth i+1:

Vi(s’) ]

Q-Value Iteration

Initialize each q-state: Q0(s,a) = 0

Repeat

For all q-states, s,a

Compute Qi+1(s,a) from Qi by Bellman backup at s,a.

Until maxs,a |Qi+1(s,a) – Qi(s,a)| <

Vi(s’) ]

Reinforcement Learning

a

s, a

s,a,s’

s’

- Markov decision processes:
- States S
- Actions A
- Transitions T(s,a,s’) aka P(s’|s,a)
- Rewards R(s,a,s’) (and discount )
- Start state s0 (or distribution P0)

- Algorithms
- Q-value iteration Q-learning

Recap: Sampling Expectations

- Want to compute an expectation weighted by P(x):
- Model-based: estimate P(x) from samples, compute expectation
- Model-free: estimate expectation directly from samples
- Why does this work? Because samples appear with the right frequencies!

Recap: Exp. Moving Average

- Exponential moving average
- Makes recent samples more important
- Forgets about the past (distant past values were wrong anyway)
- Easy to compute from the running average

- Decreasing learning rate can give converging averages

Q-Learning Update

- Q-Learning = sample-based Q-value iteration
- How learn Q*(s,a) values?
- Receive a sample (s,a,s’,r)
- Consider your old estimate:
- Consider your new sample estimate:
- Incorporate the new estimate into a running average:

Q-Learning Update

- Alternatively….

Q-Learning = sample-based Q-value iteration

- How learn Q*(s,a) values?
- Receive a sample (s,a,s’,r)
- Consider your old estimate:
- Consider your new sample estimate:
- Incorporate the new estimate into a running average:

difference = sample – Q(s, a)

Exploration / Exploitation

vs.

- greedy
- Every time step, flip a coin: with probability , act randomly
- With probability 1- , act according to current policy

- Exploration function
- Explore areas whose badness is not (yet) established
- Takes a value estimate and a count, and returns an optimistic utility, e.g. (exact form not important)
- Exploration policy π(s’)=

Q-Learning Final Solution

- Q-learning produces tables of q-values:

Q-Learning: Greedy

Q-Learning Properties

- Amazing result: Q-learning converges to optimal policy
- If you explore enough
- If you make the learning rate small enough
- … but not decrease it too quickly!
- Not too sensitive to how you select actions (!)

- Neat property: off-policy learning
- learn optimal policy without following it (some caveats)

Q-Learning – Small Problem

- Doesn’t work
- In realistic situations, we can’t possibly learn about every single state!
- Too many states to visit them all in training
- Too many states to hold the q-tables in memory

- Instead, we need to generalize:
- Learn about a few states from experience
- Generalize that experience to new, similar states
(Fundamental idea in machine learning)

Example: Pacman

- Let’s say we discover through experience that this state is bad:

- In naïve Q learning, we know nothing about related states and their Q values:

- Or even this third one!

Feature-Based Representations

- Solution: describe a state using a vector of features (properties)
- Features are functions from states to real numbers (often 0/1) that capture important properties of the state

- Example features:
- Distance to closest ghost
- Distance to closest dot
- Number of ghosts
- 1 / (dist to dot)2
- Is Pacman in a tunnel? (0/1)
- …… etc.

- Can also describe a q-state (s, a) with features (e.g. action moves closer to food)

Linear Feature Functions

- Using a feature representation, we can write a q function (or value function) for any state using a linear combination of a few weights:

- Advantage:

- Advantage: our experience is summed up in a few powerful numbers

|S|2|A| ? |S||A| ?

- Disadvantage: states may share features but actually be very different in value!

- Disadvantage:

Function Approximation

- Q-learning with linear q-functions:
- Intuitive interpretation:
- Adjust weights of active features
- E.g. if something unexpectedly bad happens, disprefer all states with that state’s features

- Formal justification: online least squares

Exact Q’s

Approximate Q’s

Q-learning with Linear Approximators

- Start with initial parameter values
- Take action a according to an explore/exploit policy(should converge to greedy policy, i.e. GLIE) transitioning from s to s’
- Perform TD update for each parameter
- Goto 2

- Q-learning can diverge. Converges under some conditions.

Minimizing Error

Imagine we had only one point x with features f(x):

Approximate q update:

“target”

“prediction”

Recap: Linear Function Approximation

- Define a set of state features f1(s), …, fn(s)
- The features are used as our representation of states
- States with similar feature values will be considered to be similar

- Often represent V(s)with linear approximation
- Approx. accuracy is fundamentally limited by the features
- Can we always define features for perfect approximation?
- Yes! Assign each state an indicator feature. (I.e. i’th feature is 1 iffi’th state is present and i represents value of i’th state)
- No! This requires far too many features, gives no generalization.

Example

10

- Grid with no obstacles, deterministic actions U/D/L/R, no discounting, -1 reward everywhere except +10 at goal
- Features for state s=(x,y): f1(s)=x, f2(s)=y (just 2 features)
- V(s) = 0 + 1 x + 2 y
- Is there a good linear approximation?
- Yes.
- 0 =10, 1 = -1, 2 = -1
- (note upper right is origin)

- V(s) = 10 - x - ysubtracts Manhattan dist.from goal reward

6

0

0

6

But What If We Change Reward?

10

- V(s) = 0 + 1 x + 2 y
- Is there a good linear approximation?
- No.

0

0

But What If…

10

+ 3 z

- Include new feature z
- z= |3-x| + |3-y|
- z is dist. to goal location

- Does this allow a good linear approx?
- 0 =10, 1 = 2 = 0, 0 = -1

- V(s) = 0 + 1 x + 2 y

0

3

0

3

RL via Policy Gradient Search

- So far all of our RL techniques have tried to learn an exact or approximate utility function or Q-function
- Learn optimal “value” of being in a state, or taking an action from state.

- Value functions can often be much more complex to represent than the corresponding policy
- Do we really care about knowing Q(s,left) = 0.3554, Q(s,right) = 0.533
- Or just that “right is better than left in state s”

- Motivates searching directly in a parameterized policy space
- Bypass learning value function and “directly” optimize the value of a policy

Aside: Gradient Ascent

- Gradient ascent iteratively follows the gradient direction starting at some initial point
- Initialize to a random value
- Repeat until stopping condition
where

Local optima of f

With proper decay of learning rate gradient descent is guaranteed to converge to local optima.

2

1

RL via Policy Gradient Ascent

- The policy gradient approach has the following schema:
- Select a space of parameterized policies
- Compute the gradient of the value of current policy wrt parameters
- Move parameters in the direction of the gradient
- Repeat these steps until we reach a local maxima
- Possibly also add in tricks for dealing with bad local maxima (e.g. random restarts)

- So we must answer the following questions:
- How should we represent parameterized policies?
- How can we compute the gradient?

Parameterized Policies

- One example of a space of parametric policies is:
where may be a linear function, e.g.

- The goal is to learn parameters that give a good policy
- Note that it is not important that be close to the actual Q-function
- Rather we only require is good at ranking actions in order of goodness

Policy Gradient Ascent

- Policy gradient ascent tells us to iteratively update parameters via:
- Problem: ()is generally very complex and it is rare that we can compute a closed form for the gradient of () even if we have an exact model of the system.
- Key idea: estimate the gradient based on experience

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